# Simplified and Space-Optimal Semi-Streaming for $(2+\epsilon)$-Approximate Matching

## TL;DR

This paper simplifies and improves the analysis of a semi-streaming algorithm for approximate maximum weight matching, reducing space complexity to the optimal bound while maintaining a good approximation ratio.

## Contribution

It provides two simplified, intuitive analyses of an existing algorithm, achieving optimal space complexity of O(n log n) bits for (2+ε)-approximate matching.

## Key findings

- Achieved optimal space complexity of O(n log n) bits.
- Provided simplified, intuitive primal-dual based analyses.
- Maintained a (2+ε)-approximation ratio.

## Abstract

In a recent breakthrough, Paz and Schwartzman (SODA'17) presented a single-pass ($2+\epsilon$)-approximation algorithm for the maximum weight matching problem in the semi-streaming model. Their algorithm uses $O(n\log^2 n)$ bits of space, for any constant $\epsilon>0$. We present two simplified and more intuitive analyses, for essentially the same algorithm, which also improve the space complexity to the optimal bound of $O(n\log n)$ bits --- this is optimal as the output matching requires $\Omega(n\log n)$ bits. Our analyses rely on a simple use of the primal-dual method and a simple accounting method.